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Elastic Properties of Solids Topics Discussed in Kittel, Ch. 3, pages 73-85. Hooke's “Law”.

Elastic Properties of Solids Topics Discussed in Kittel, Ch. 3, pages 73-85

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A property of an ideal spring of spring constant k is that it takes twice as much force to stretch the spring twice as far. That is, if it is stretched a distance x, the restoring force is given by F = - kx. The spring is then said to obey

Hooke's “Law”.

- An elastic medium is one in which a disturbance can be analyzed in terms of Hooke’s “Law” forces.
- Consider the propagation of a mechanical wave
- (disturbance) in a solid.
- We are interested in the case of very long wavelengths, when the wavelength is much, much larger than the interatomic spacing:
- >> a
- so that the solid can be treated as a continuous elastic medium & the fact that there are atoms on a lattice is irrelevant to the wave propagation.

A Prototype Hooke’s Law System

A Mass-Spring System

in which a mass m is

attached to an ideal

spring of spring

constant k. That is, the

Simple Harmonic

Oscillator

(SHO)

Simple Harmonic Oscillator

Stretch the spring a distance A & release it:

Fig. 1

Fig. 2

Fig. 3

- In the absence of friction, the oscillations go on forever.
- The Newton’s 2nd Law equation of motionis:
- F = ma = m(d2x/dt2) = -kx
- Define:(ω0)2 k/m (d2x/dt2) + (ω0)2x = 0
- A standard 2nd order time dependent differential equation!

∑Fx= 0 = mg - kx0

orx0 = (mg/k)

Newton’s 2nd Law

Equation of Motion:

This is the same as

before, but the

equilibrium position is

x0instead of x = 0

Simple Harmonic Oscillator

Hooke’s “Law” for a vertical spring (take + x as down):

Static Equilibrium:

An Elastic Medium is defined to be one in which a disturbance from equilibrium obeys Hooke’s “Law”so that a local deformation is proportional to an applied force.

If the applied force gets too large, Hooke’s “Law” no longer holds. If that happens the medium is no longer elastic. This is called the Elastic Limit.

The Elastic Limit is the point at which permanent deformation occurs, that is, if the force is taken off the medium, it will not return to its original size and shape.

- Sound waves are mechanical waves which propagate through a material medium (solid, liquid, or gas) at a speed which depends on the elastic & inertial properties of the medium. There are 2 types of wave motion for sound waves:
Longitudinal

and

Transverse

Longitudinal Waves

- Because we are considering only long wavelength mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.

Longitudinal Waves

Transverse Waves

- Because we are considering only long wavelength mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.

- Sound waves propagate through solids. This tells us that wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The detailed atomic structure is unimportant for these waves &their propagation is governed by the macroscopic elastic propertiesof the crystal.
- So, the reason for discussingsound waves is that
theycorrespond to the low frequency, long

wavelength limitof the more general lattice

vibrations we have been considering up to now.

- At a given frequency and in a given direction in a crystal it is possible to transmit 3 different kinds of sound waves, differing in their direction of polarizationand in general also in their velocity.

Consider Longitudinal Elastic Wave

Propagation in a Solid Bar

- At the point xthe elastic displacement (or change in length) is U(x)&the strain e is defined as the change in length per unit length.

A

x x+dx

So, consider sound waves propagating in a solid, when theirwavelength is very long, so that the solid may be treated as a continous medium. Such waves are referred to as elastic waves.

In general, a Stress Sat a point in space is defined as the force per unit area at that point.

x x+dx

- Hooke’s“Law”tells us that, at point x & time t in the bar, the stress Sproduced by an elastic wave propagation is proportional to the strain e.That is:

A

C Young’sModulus

To analyze the dynamics of the bar, choose an arbitrary segment of length dx as shown above. UseNewton’s 2nd Law to write for the motion of this segment,

C Young’sModulus

A

x x+dx

Mass Acceleration = Net Force resulting from stress

Equation of Motion

So, this becomes:

Cancelling common terms inAdx gives:

This is the wave equation a plane

wave solution which gives the

sound velocityvs:

Plane wave solution:

k = wave number = (2π/λ), ω = frequency,A = amplitude

Unlike the case for the discrete lattice, thedispersion relation ω(k)in this long wavelength limitis the simple equation:

ω

Continuum

0

Discrete

k

- At small λ (k → ∞), scattering from discrete atoms occurs.
- At long λ(k → 0),(continuum) no scattering occurs.
- When k increases the sound velocity decreases.
- As k increases further, the scattering becomes greater since the strength of scattering increases as the wavelength decreases, and the velocity decreases even further.

The speed VL with which a longitudinal elastic wave moves through a medium of density ρis given by:

C Bulk Modulus

ρ Mass Density

- The velocity of sound is in general a function of the direction of propagation in crystalline materials.
- Solids will sustain the propagation of transverse waves, which travel more slowly than longitudinal waves.
- The larger the elastic modulus & the smaller the density, the larger the sound speed is.

Speed of Sound for Several Common Solids

Most calculated VL values are in reasonable agreement with measurements. Sound speeds are of the order of 5000 m/s in typical metallic, covalent & ionic solids :